In UV map can't move shared vertices independently
This happens because you have the "Keep UV and edit mode mesh selection in sync" button enabled, the selected vertex in the UV editor, is on a seam, so in 3D view, it is shared by 4 faces - assuming you are working with quads- so when selected with the sync option enabled, it will select all related coordinates in the UV editor, disable it by clicking the button, in the image below, it's the button in the lower right corner.
As you can see, I have it enabled, here.



Now check the image below, I disabled the Sync button.



Hope this helps, happy blending.

Edit:

When you have the Sync button disabled, you can only see in the UV Editor, the faces that you have it selected in the 3D View in Edit Mode, so make sure you select all the faces you need to have it available before unchecking the Sync button, or use the 3D View to select the faces/edges/vertices you need, and tweak them in the UV Editor with the Sync button unchecked.